Administrators at Texas A&M University were interested in estimating the percentage of students who are the first in their family to go to college. The A&M student body has about 46,000 members. The president of the university plans a speech to an alumni group. He plans to talk about the proportion of students who responded in the survey that they are the first in their family to attend college, but the first draft of his speech treats that proportion as the actual proportion of current A&M students who are the first in their families to attend college. Explain to the president the difference between the proportion of respondents who are first attenders and the proportion of the entire student body that are first attenders. Use appropriate statistics terminology.

A professor teaching a large lecture class of 350 students samples her class by rolling a die. Then, starting with the row number on the die (1 to 6), she passes out a survey to every fourth row of the large lecture hall. She says that this is a Simple Random Sample because everyone had an equal opportunity to sit in any seat and because she randomized the choice of rows. What do you think? Be specific.

Find the indicated probabilities using the Poisson distribution. If convenient, use a Poisson probability table or technology tool to find the probabilities.

The mean number of days with 0.01 inch or more precipitation per month for Lewistown, Idaho, is about 8.7. Find the probability that in a given month a) there are exactly 9 days with 0.01 inch or more precipitation, b) there are at most 9 days with 0.01 inch or more precipitation, and c) there are more than 9 days with 0.01 inch or more precipitation.

The data below gives the MPG ratings for cars owned by 15 Placid High School seniors. Find the following statistics about the MPG ratings. 15.9, 17.8, 21.6, 25.2, 31.1, 29, 28.6, 32, 34, 14, 19.8, 19.5, 20.1, 27.7, 25.5 a. mean b. median c. median of the lowest seven scores d. lower quartile, $Q_1$ e. median of the highest seven scores f. upper quartile, $\mathrm{Q}_{3}$ g. interquartile range h. range i. boundary for the upper outliers j. boundary for the lower outliers k. How many outliers are in this data set?

Let $f(x)=e^{-A x^2 / 2}$, where $A>0$. Given any $n$ numbers $a_1, a_2, \ldots, a_n$, set

$$\Phi(x)=f\left(x-a_1\right) f\left(x-a_2\right) \cdots f\left(x-a_n\right)$$

(a) Assume $n=2$ and prove that $\Phi(x)$ attains its maximum value at the average $x=\frac{1}{2}\left(a_1+a_2\right)$. Hint: Show that $d / d x \ln (f(x))=-A x$ and calculate $\Phi^{\prime}(x)$ using logarithmic differentiation.

(b) Show that for any $n, \Phi(x)$ attains its maximum value at $x=$ $\frac{1}{n}\left(a_1+a_2+\cdots+a_n\right)$. This fact is related to the role of $f(x)$ (whose graph is a bell-shaped curve) in statistics.

Tennis balls are packaged in a cylindrical can, and since they're round, that leaves a lot of empty space in the can. How much space? The radius of a tennis ball is $3.3 \mathrm{~cm}$; the can has the same radius for a nice tight fit, and the height of the can is six times that radius, which is $19.8 \mathrm{~cm}$.

Use the unit converter website you found in the Technology portion of the portfolio page to convert this volume from cubic centimeters to liters.

When it is operating correctly, a machine for manufacturing tennis balls produces balls with a mean weight of 57 .6 grams. The last eight balls manufactured had weights

57.3, 57.4, 57.2, 57.5, 57.4, 57.1, 57.3, 57.0

a. Using a calculator or software, find the test statistic and P-value for a test of whether the process is in control against the alternative that the true mean of the process now differs from 57.6 .

b. For a significance level of 0.05, explain what you would conclude. Express your conclusion so it would be understood by someone who never studied statistics.

c. If your decision in part bis in error, what type of error have you made?

Why do many people oppose regulations and restrictions designed to cut greenhouse gas emissions?

a. Scientific uncertainty in a technological age undermines efforts to curb the effects of the technology.

b. The costs of taking action are immediate, but the benefits may not be felt for decades.

c. The technologies necessary to accomplish the cuts have not been invented yet.

d. They believe the Environmental Protection Agency standards are an example of government overreach.

e. They do not want to change the way they operate their businesses or industries.

The table shows examples of the expected return on an investment in a new company. Use technology to find a quadratic function that is a reasonable fit to the data. Use the function to predict the return on an investment of $\$ 4000$. Round to the nearest thousand dollars.

$$\begin{array}{|l|c|c|c|c|c|c|} \hline Investment (\$) & 50 & 100 & 200 & 500 & 1000 & 1500 \\ \hline Return (\$) & 2 & 12 & 50 & 250 & 1000 & 2000 \\ \hline \end{array}$$

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Consider full IQ scores for a random sample of subjects with low lead levels in their blood and another random sample of subj ects with high lead levels in their blood. The statistics are summarized below.

$$\begin{array} { l l } { \text { Low Blood Lead Level: } } & { n = 78 , \overline { x } = 92.88462 , s = 15.34451 } \\ { \text { High Blood Lead Level: } } & { n = 21 , \overline { x } = 86.90476 , s = 8.988352 } \end{array}$$

Does exposure to lead appear to have an effect on IQ scores?

In a USA Today survey of 510 people, 53% said that we should replace passwords with biometric security, such as fingerprints. The accompanying Statdisk data results from a test of the claim that half of us say that we should replace passwords with biometric security.

$$\begin{matrix} \text{Sample proportion:} & \text{0.5294118}\\ \text{Test Statistics, z:} & \text{1.3284}\\ \text{Critical z:} & \text{$\pm$1.9600}\\ \text{P-Value:} & \text{0.1840}\\ \end{matrix}$$

				Use the normal distribution as an approximation to the binomial distribution. Use a 0.05 significance level and answer the following: Is the test two-tailed, left-tailed, or right-tailed?

Over the past fifty years the average American's leisure time has increased by between 4 and 8 hours a week. Some economists think that this increase is primarily driven by a rise in wage rates. a. Use the income and substitution effects to describe the labor supply for the average American. Which effect dominates? b. In addition to increasing wages, a study by the Bureau of Labor Statistics finds labor force participation for women is projected to steadily increase through 2024. For the average woman who has entered the labor force, which c. Draw typical individual labor supply curves that illustrate your answers to part $a$ and part b above.

Suppose n = 36 observations are taken from a normal distribution where $\sigma=8.0$ for the purpose of testing $H_{0}: \mu=60$ versus $H_{1}: \mu \neq 60$ at the $\alpha=0.07$ level of significance. The lead investigator skipped statistics class the day decision rules were being discussed and intends to reject $H_{0}$ if $\bar{y}$ falls in the region $\left(60-\bar{y}^{*}, 60+\bar{y}^{*}\right)$. (a) Find $\bar{y}^{*}$ (b) What is the power of the test when $\mu=62 ?$ (c) What would the power of the test be when $\mu=62$ if the critical region had been defined the correct way?

A new production technology for making vitamins is invented by a college professor who decides not to patent it. Thus, it is available for anybody to copy and put into use. The TC per bottle for production up to $100,000$ bottles per day is given in the following table.

$$\begin{array}{rr} \text { Output } & \text { TC } \\ \hline 25,000 & \$ 50,000 \\ 50,000 & 70,000 \\ 75,000 & 75,000 \\ 100,000 & 80,000 \end{array}$$

Do both production technologies show constant returns to scale?